NAG Toolbox: nag_sparse_direct_real_gen_refine (f11mh)

Purpose

nag_sparse_direct_real_gen_refine (f11mh) returns error bounds for the solution of a real sparse system of linear equations with multiple right-hand sides, AX = BAX=B or ATX = BATX=B. It improves the solution by iterative refinement in standard precision, in order to reduce the backward error as much as possible.

Description

nag_sparse_direct_real_gen_refine (f11mh) returns the backward errors and estimated bounds on the forward errors for the solution of a real system of linear equations with multiple right-hand sides AX = BAX=B or ATX = BATX=B. The function handles each right-hand side vector (stored as a column of the matrix BB) independently, so we describe the function of nag_sparse_direct_real_gen_refine (f11mh) in terms of a single right-hand side bb and solution xx.

Given a computed solution xx, the function computes the component-wise backward errorββ. This is the size of the smallest relative perturbation in each element of AA and bb such that if xx is the exact solution of a perturbed system:

(A + δA)x = b + δb

then |δaij| ≤ β|aij| and |δbi| ≤ β|bi|.

(A+δA)x=b+δbthen |δaij|≤β|aij| and |δbi|≤β|bi|.

Then the function estimates a bound for the component-wiseforward error in the computed solution, defined by: